85.41.2 problem 2 (a)

Internal problem ID [22768]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. C Exercises at page 178
Problem number : 2 (a)
Date solved : Thursday, October 02, 2025 at 09:14:27 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \ln \left (x -1\right ) x}{2}-\frac {c_2 \ln \left (x +1\right ) x}{2}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 33

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],{x,1}]+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 x-\frac {1}{2} c_2 (x \log (1-x)-x \log (x+1)+2) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False